Bipolar Complex Fuzzy Rough Sets and Their Applications in Multicriteria Decision Making

Bipolar complex fuzzy set (BCFS) is a more advanced and powerful phenomenon as it consists of two-dimensional data with positive and negative impacts of an element. It can solve the data consisting of the positive and negative impacts of an element which is a bipolar fuzzy set (BFS). It also covers the two-dimensional complex data which is a complex fuzzy set (CFS). Due to these attributes, BFS and CFS are less useful in comparison with BCFS to capture vagueness, complexity, and ambiguity in the data. Furthermore, lower and upper approximations based on equivalency relations constitute another significant phenomenon known as rough set (RS). This structure is also more powerful in dealing with real-life dilemmas. Rather than comparing the RS and BCFS, we combine both phenomena to handle the complexity more powerfully to deal with such types of phenomena that are not handled by other structures. So, by combining both phenomena, we introduce a novel structure known to be bipolar complex fuzzy rough set (BCFRS) in this manuscript. After that, we define some important operations, some significant properties related to this structure, and some aggregation operators (AOs) to solve decision-making (DM) problems related to cyber security. We address a practical application of cyber security (C-S) in computing for the protection of critical data to demonstrate the usefulness of the multi-attribute DM(MADM) approach. Based on the various criteria and attributes given by the experts, we find the best and better alternative to the C-S by applying the MADM approach. We get the A4 as the best and finest alternative by using bipolar complex fuzzy rough (BCFR) weighted arithmetic averaging (BCFRWAA), BCFR ordered weighted arithmetic averaging (BCFROWAA), and BCFR ordered weighted geometric averaging (BCFROWGA) operators. And, by using BCFR weighted geometric averaging (BCFRWGA), we get the A3 as the finest alternative. Lastly, to prove the superiority, validity, and generalization of our unique established theory, we give a detailed comparative study of our established work with several prevalent theories.